THE GREEKS HAD A NAME FOR IT 165

Conjugacy Classes

FACT: Any group H splits up into subsets C with the

following two properties:

1. Each subset C is obtained in the following way: You take

any group element, say x. Then you take all of the elements of

the group, call them g’s, and form the group products

g ◦ x ◦ g

−1

. Notice that x itself is one of these products, because

e ◦ x ◦ e

−1

= x (where e, as usual, is the neutral element). The

subset that consists of all these g ◦ x ◦ g

−1

’s is one of the C’s.

For example, x is an element of the C that you get by starting

with x. It does not matter which x in C you start with. You get

the same bunch of elements, namely C!

2. Any two elements of C have the same character value

under every representation r. In symbols: For any matrix

representation of the group H, call it r,ifx and y are in the

same C, then χ

r

(x) = χ

r

(y).

These subsets C are called the “conjugacy classes of H.” The C’s

are often quite large. The larger they are, the more the group law of

H fails to be commutative. At any rate, in a commutative group,

each conjugacy class contains only one element. This is because

if the group law is commutative, then g ◦ x ◦ g

−1

= g ◦ g

−1

◦ x =

e ◦ x = x.

The deﬁnition of the C’s is actually accomplished just by prop-

erty 1. There is only one way of splitting up H into C’s that possess

property 1. The proof is just an exercise using the axioms of the

group law. Then property 2 is not too difﬁcult to prove if you

know some linear algebra. The really amazing fact—and it is a

fact about representation theory—is that if x is in C and y is in

C

, which is some other conjugacy class, then there is some matrix

representation r such that χ

r

(x) = χ

r

(y). We will give an example of

this fact a little later for the group

{1,2,3}

.

Even more amazing is the important role that the characters

play in the number theory of Galois groups. Before we get to

that, though, we have some more to explain about group represen-

tations and their characters.

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